The effect of a soap film on a catenary: Measurement of surface tension from the triangular configuration

A chain assumes the well-known shape known as a catenary when it hangs loosely from two points in a gravitational field. The correct solution of the catenary was one of the early triumphs of the newly invented calculus of variations at the end of the 17th century. Here we revisit the catenary and show that, for a chain hanging from a horizontal rod, three new and distinct configurations are possible if a soap film covers the area bounded by the chain and the rod. We first review the general problem and discuss the conditions under which the chain assumes a concave, triangular or convex configuration. The deciding factor is the strength of surface tension relative to the gravitational force per unit length of the chain. The conditions under which the chain assumes the shape of a perfect triangle are discussed in greater detail and analysed to obtain the tension along the chain. The triangular configuration is especially intriguing to undergraduates and may be used as a simple experiment to obtain the surface tension of the soap solution by measuring just one angle of the triangle.